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Properties

Label 2-349-349.123-c1-0-24
Degree $2$
Conductor $349$
Sign $-0.327 + 0.944i$
Analytic cond. $2.78677$
Root an. cond. $1.66936$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0377 + 0.0217i)2-s + (1.71 − 2.97i)3-s + (−0.999 + 1.73i)4-s + (1.73 − 3.00i)5-s + 0.149i·6-s + (−2.79 + 1.61i)7-s − 0.174i·8-s + (−4.39 − 7.60i)9-s + 0.151i·10-s + 0.153i·11-s + (3.42 + 5.93i)12-s + (2.06 − 1.19i)13-s + (0.0704 − 0.122i)14-s + (−5.96 − 10.3i)15-s + (−1.99 − 3.45i)16-s + 5.10·17-s + ⋯
L(s)  = 1  + (−0.0266 + 0.0154i)2-s + (0.990 − 1.71i)3-s + (−0.499 + 0.865i)4-s + (0.777 − 1.34i)5-s + 0.0610i·6-s + (−1.05 + 0.610i)7-s − 0.0616i·8-s + (−1.46 − 2.53i)9-s + 0.0479i·10-s + 0.0462i·11-s + (0.989 + 1.71i)12-s + (0.571 − 0.330i)13-s + (0.0188 − 0.0326i)14-s + (−1.54 − 2.66i)15-s + (−0.498 − 0.863i)16-s + 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-0.327 + 0.944i$
Analytic conductor: \(2.78677\)
Root analytic conductor: \(1.66936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 349,\ (\ :1/2),\ -0.327 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923649 - 1.29758i\)
\(L(\frac12)\) \(\approx\) \(0.923649 - 1.29758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (-13.4 - 12.9i)T \)
good2 \( 1 + (0.0377 - 0.0217i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.71 + 2.97i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.73 + 3.00i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.79 - 1.61i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.153iT - 11T^{2} \)
13 \( 1 + (-2.06 + 1.19i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.10T + 17T^{2} \)
19 \( 1 + (2.56 - 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.19 - 3.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.29 + 3.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.19T + 31T^{2} \)
37 \( 1 + 1.35T + 37T^{2} \)
41 \( 1 - 7.68T + 41T^{2} \)
43 \( 1 + (-1.29 - 0.749i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.44iT - 47T^{2} \)
53 \( 1 - 6.29iT - 53T^{2} \)
59 \( 1 + (7.84 + 4.53i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.70iT - 61T^{2} \)
67 \( 1 - 5.98T + 67T^{2} \)
71 \( 1 + (-8.67 + 5.00i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.31 + 4.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.12iT - 79T^{2} \)
83 \( 1 + (-2.69 - 4.65i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.728 + 0.420i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.5 - 7.80i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.01447491382790622084912651092, −9.623673102788096577941831554549, −9.165529003361908449265326288786, −8.285311758525996095523128167227, −7.80811068005917915767211501986, −6.41404627848832845665811212167, −5.63175513060945874573395937441, −3.66301627484226408824793619268, −2.60016348284967469538617951538, −1.10045779222773198679560362829, 2.63257453195105007075496621233, 3.50553371107388202058264374204, 4.61623250592407072015809853820, 5.83697967880226438267981549963, 6.82667861827959994232874422534, 8.453935985564638637329545561313, 9.477959436710969881721332004486, 9.828518993361465845029979477388, 10.68717311495413339734514768834, 10.86947955700627276655777292776

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