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Properties

Label 1-349-349.9-r0-0-0
Degree $1$
Conductor $349$
Sign $0.853 + 0.521i$
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  + (0.590 + 0.806i)2-s + (0.750 − 0.661i)3-s + (−0.302 + 0.953i)4-s + (0.874 − 0.484i)5-s + (0.976 + 0.214i)6-s + (0.997 + 0.0721i)7-s + (−0.947 + 0.319i)8-s + (0.126 − 0.992i)9-s + (0.907 + 0.419i)10-s + (0.647 + 0.762i)11-s + (0.403 + 0.915i)12-s + (−0.999 + 0.0361i)13-s + (0.530 + 0.847i)14-s + (0.336 − 0.941i)15-s + (−0.817 − 0.576i)16-s + (−0.947 − 0.319i)17-s + ⋯
L(s)  = 1  + (0.590 + 0.806i)2-s + (0.750 − 0.661i)3-s + (−0.302 + 0.953i)4-s + (0.874 − 0.484i)5-s + (0.976 + 0.214i)6-s + (0.997 + 0.0721i)7-s + (−0.947 + 0.319i)8-s + (0.126 − 0.992i)9-s + (0.907 + 0.419i)10-s + (0.647 + 0.762i)11-s + (0.403 + 0.915i)12-s + (−0.999 + 0.0361i)13-s + (0.530 + 0.847i)14-s + (0.336 − 0.941i)15-s + (−0.817 − 0.576i)16-s + (−0.947 − 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.853 + 0.521i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.853 + 0.521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.482007260 + 0.6983996065i\)
\(L(\frac12)\) \(\approx\) \(2.482007260 + 0.6983996065i\)
\(L(1)\) \(\approx\) \(1.935726807 + 0.4463425180i\)
\(L(1)\) \(\approx\) \(1.935726807 + 0.4463425180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.590 + 0.806i)T \)
3 \( 1 + (0.750 - 0.661i)T \)
5 \( 1 + (0.874 - 0.484i)T \)
7 \( 1 + (0.997 + 0.0721i)T \)
11 \( 1 + (0.647 + 0.762i)T \)
13 \( 1 + (-0.999 + 0.0361i)T \)
17 \( 1 + (-0.947 - 0.319i)T \)
19 \( 1 + (0.837 + 0.546i)T \)
23 \( 1 + (-0.999 + 0.0361i)T \)
29 \( 1 + (0.403 - 0.915i)T \)
31 \( 1 + (-0.370 + 0.928i)T \)
37 \( 1 + (0.468 - 0.883i)T \)
41 \( 1 + (-0.947 + 0.319i)T \)
43 \( 1 + (-0.983 + 0.179i)T \)
47 \( 1 + (0.267 + 0.963i)T \)
53 \( 1 + (-0.161 + 0.986i)T \)
59 \( 1 + (0.750 - 0.661i)T \)
61 \( 1 + (-0.994 - 0.108i)T \)
67 \( 1 + (0.0541 + 0.998i)T \)
71 \( 1 + (0.750 + 0.661i)T \)
73 \( 1 + (-0.436 - 0.899i)T \)
79 \( 1 + (-0.161 - 0.986i)T \)
83 \( 1 + (-0.0180 + 0.999i)T \)
89 \( 1 + (-0.232 - 0.972i)T \)
97 \( 1 + (0.874 + 0.484i)T \)
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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

−24.42678388223301598384584996174, −24.202276950472388791715904838157, −22.41303522734557468749514365441, −21.84412144885764025000782544604, −21.53114262652517713732263411181, −20.279127454977713000209705058293, −19.91428350499475758461281822682, −18.71323156522293248998450667171, −17.85738460216248399855251972878, −16.75648048550715098161153634579, −15.2657522648823444199963335898, −14.658238501745047882764603502634, −13.8933245516985778671638673933, −13.38592127215472857598285985389, −11.82384037011906471674612935344, −10.99855384594559616180427868993, −10.147865360796399220001312349290, −9.344067951447297769401972503017, −8.397883380074267208496460876398, −6.85067100108692382728156749696, −5.492093536858222297076966389098, −4.6940470110153037877002555346, −3.56988109019594695893129809540, −2.498423583027812157197807731005, −1.67544306135465201393845855140, 1.65277237273616200990875556914, 2.56690358671523784682711739288, 4.1754477614545697710017338392, 5.04266504045593092859794881263, 6.210221633661644789620955668625, 7.19844602692739506927201108761, 8.02985972044350338769392029456, 8.99445552597335938405496491985, 9.788121972037686661812015633589, 11.815356329233795112633760719037, 12.37652231005486615751033666693, 13.43658349426316494720517206131, 14.21098401958136896711163632371, 14.653708903549584141889582766565, 15.764449438730874871333778757581, 17.08126917227116115769070890939, 17.71956172937474632653335551498, 18.262237127521570058985182575416, 19.95582456730507126246393571294, 20.49401558919707182295635644142, 21.50112339649795849794194070911, 22.23613130664218378788546519907, 23.476774867792731564067459823509, 24.36950445655239440659139473526, 24.86305516237719047073957735699

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