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Properties

Label 1-349-349.81-r0-0-0
Degree $1$
Conductor $349$
Sign $0.973 - 0.227i$
Analytic cond. $1.62074$
Root an. cond. $1.62074$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.302 + 0.953i)2-s + (0.126 − 0.992i)3-s + (−0.817 − 0.576i)4-s + (0.530 − 0.847i)5-s + (0.907 + 0.419i)6-s + (0.989 + 0.143i)7-s + (0.796 − 0.605i)8-s + (−0.968 − 0.250i)9-s + (0.647 + 0.762i)10-s + (−0.161 + 0.986i)11-s + (−0.674 + 0.738i)12-s + (0.997 − 0.0721i)13-s + (−0.436 + 0.899i)14-s + (−0.773 − 0.633i)15-s + (0.336 + 0.941i)16-s + (0.796 + 0.605i)17-s + ⋯
L(s)  = 1  + (−0.302 + 0.953i)2-s + (0.126 − 0.992i)3-s + (−0.817 − 0.576i)4-s + (0.530 − 0.847i)5-s + (0.907 + 0.419i)6-s + (0.989 + 0.143i)7-s + (0.796 − 0.605i)8-s + (−0.968 − 0.250i)9-s + (0.647 + 0.762i)10-s + (−0.161 + 0.986i)11-s + (−0.674 + 0.738i)12-s + (0.997 − 0.0721i)13-s + (−0.436 + 0.899i)14-s + (−0.773 − 0.633i)15-s + (0.336 + 0.941i)16-s + (0.796 + 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.319329728 - 0.1519811721i\)
\(L(\frac12)\) \(\approx\) \(1.319329728 - 0.1519811721i\)
\(L(1)\) \(\approx\) \(1.099728520 + 0.002183692871i\)
\(L(1)\) \(\approx\) \(1.099728520 + 0.002183692871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.302 + 0.953i)T \)
3 \( 1 + (0.126 - 0.992i)T \)
5 \( 1 + (0.530 - 0.847i)T \)
7 \( 1 + (0.989 + 0.143i)T \)
11 \( 1 + (-0.161 + 0.986i)T \)
13 \( 1 + (0.997 - 0.0721i)T \)
17 \( 1 + (0.796 + 0.605i)T \)
19 \( 1 + (0.403 + 0.915i)T \)
23 \( 1 + (0.997 - 0.0721i)T \)
29 \( 1 + (-0.674 - 0.738i)T \)
31 \( 1 + (-0.725 - 0.687i)T \)
37 \( 1 + (-0.561 - 0.827i)T \)
41 \( 1 + (0.796 - 0.605i)T \)
43 \( 1 + (0.935 - 0.353i)T \)
47 \( 1 + (-0.856 + 0.515i)T \)
53 \( 1 + (-0.947 - 0.319i)T \)
59 \( 1 + (0.126 - 0.992i)T \)
61 \( 1 + (0.976 + 0.214i)T \)
67 \( 1 + (-0.994 + 0.108i)T \)
71 \( 1 + (0.126 + 0.992i)T \)
73 \( 1 + (-0.619 + 0.785i)T \)
79 \( 1 + (-0.947 + 0.319i)T \)
83 \( 1 + (-0.999 - 0.0361i)T \)
89 \( 1 + (-0.891 + 0.452i)T \)
97 \( 1 + (0.530 + 0.847i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.269244964467312709632049551515, −23.715313138895553005810708792482, −22.73181391014012181898714037329, −21.949996581869365874269382785664, −21.10059462963235526145877537452, −20.883190872693145130338633942995, −19.66730324774774137588218547876, −18.61037853529317165074497483418, −17.95652171949301630742808485040, −16.997161299553782285797084787336, −16.04923597123532684454718907654, −14.703558639439238300783783806464, −14.04541665101461497993654967089, −13.28596036444893430880432217771, −11.48826410852006237374751694083, −11.08847833087429682661648933452, −10.43934302846309169072260475107, −9.29591127791067107383194474197, −8.60992988574396225363745010327, −7.42498794933289301956910132829, −5.67170268877527591067034382300, −4.80555580840589908936349126277, −3.4327672383478946317164105189, −2.88801533897530883770531756424, −1.35256739705744824324477649721, 1.16726258784575958623033051973, 1.8912489950783573224487899641, 4.07088568859650995521388084468, 5.42022512022882781638002815354, 5.8754950109031898105604517283, 7.27558478971222007709027935384, 8.01426921749674046924425472781, 8.753505736520371374608855814158, 9.73546801803385990156080239537, 11.099520803425871361443419978987, 12.49499119242561121128749422192, 13.064330146479471480292333556164, 14.15936809984586199895787884023, 14.72596073120372843872810685927, 15.95538299711152141209843503470, 17.108219974373178417151228388797, 17.53972935019038515659953781392, 18.377696205992628630177173483398, 19.13294891835169213472979603073, 20.50361466301289488310412665013, 20.97792852457872761903548458516, 22.62612562568079023841293071913, 23.398053623431515126872911675448, 24.12269311587389427915411902758, 24.84229013163442857152210886866

Graph of the $Z$-function along the critical line