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Properties

Label 2-350-175.103-c1-0-5
Degree $2$
Conductor $350$
Sign $0.913 + 0.405i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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  + (−0.998 − 0.0523i)2-s + (0.194 + 0.239i)3-s + (0.994 + 0.104i)4-s + (2.09 + 0.786i)5-s + (−0.181 − 0.249i)6-s + (−1.59 − 2.11i)7-s + (−0.987 − 0.156i)8-s + (0.603 − 2.84i)9-s + (−2.04 − 0.894i)10-s + (3.14 − 0.668i)11-s + (0.168 + 0.258i)12-s + (0.507 + 0.258i)13-s + (1.48 + 2.19i)14-s + (0.217 + 0.654i)15-s + (0.978 + 0.207i)16-s + (−0.659 − 1.71i)17-s + ⋯
L(s)  = 1  + (−0.706 − 0.0370i)2-s + (0.112 + 0.138i)3-s + (0.497 + 0.0522i)4-s + (0.936 + 0.351i)5-s + (−0.0740 − 0.101i)6-s + (−0.603 − 0.797i)7-s + (−0.349 − 0.0553i)8-s + (0.201 − 0.947i)9-s + (−0.647 − 0.283i)10-s + (0.948 − 0.201i)11-s + (0.0485 + 0.0747i)12-s + (0.140 + 0.0717i)13-s + (0.396 + 0.585i)14-s + (0.0562 + 0.169i)15-s + (0.244 + 0.0519i)16-s + (−0.159 − 0.416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.913 + 0.405i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.913 + 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14358 - 0.242476i\)
\(L(\frac12)\) \(\approx\) \(1.14358 - 0.242476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.998 + 0.0523i)T \)
5 \( 1 + (-2.09 - 0.786i)T \)
7 \( 1 + (1.59 + 2.11i)T \)
good3 \( 1 + (-0.194 - 0.239i)T + (-0.623 + 2.93i)T^{2} \)
11 \( 1 + (-3.14 + 0.668i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (-0.507 - 0.258i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.659 + 1.71i)T + (-12.6 + 11.3i)T^{2} \)
19 \( 1 + (-0.151 - 1.43i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (0.118 - 2.26i)T + (-22.8 - 2.40i)T^{2} \)
29 \( 1 + (-4.15 + 5.72i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.85 - 4.16i)T + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (-6.45 + 4.19i)T + (15.0 - 33.8i)T^{2} \)
41 \( 1 + (-8.64 + 2.80i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-3.20 - 3.20i)T + 43iT^{2} \)
47 \( 1 + (9.10 + 3.49i)T + (34.9 + 31.4i)T^{2} \)
53 \( 1 + (7.23 - 5.86i)T + (11.0 - 51.8i)T^{2} \)
59 \( 1 + (5.10 - 5.67i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-0.562 + 0.506i)T + (6.37 - 60.6i)T^{2} \)
67 \( 1 + (5.42 - 2.08i)T + (49.7 - 44.8i)T^{2} \)
71 \( 1 + (8.84 + 6.42i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.29 - 9.69i)T + (-29.6 - 66.6i)T^{2} \)
79 \( 1 + (-3.38 - 7.60i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (-1.85 + 11.6i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-6.03 - 6.69i)T + (-9.30 + 88.5i)T^{2} \)
97 \( 1 + (-0.349 - 2.20i)T + (-92.2 + 29.9i)T^{2} \)
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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

−11.19597270084120127198905139060, −10.27760109531147228113985012668, −9.494802929584173727585430042789, −9.079149452171644534802430946281, −7.56999276964284175963420451161, −6.59025157248934306651476851710, −6.03658754519781680053883175718, −4.11957858250932266372968962564, −2.95405556000055448247513334986, −1.18362651448970390742369687286, 1.61922975316255656227003181161, 2.78009355836588842925765249891, 4.70962870058092293663332378102, 5.97971944839805580810519691707, 6.67008640783997104258053310840, 7.996265193439953381904528055320, 8.940678551475623568383241605737, 9.531646759647807745625853677696, 10.42504510331247137852893676519, 11.40733227112984648074750726845

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