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Properties

Label 2-119-17.2-c1-0-5
Degree $2$
Conductor $119$
Sign $-0.763 + 0.646i$
Analytic cond. $0.950219$
Root an. cond. $0.974792$
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  + (−1.92 − 1.92i)2-s + (2.29 − 0.949i)3-s + 5.42i·4-s + (−0.942 − 2.27i)5-s + (−6.24 − 2.58i)6-s + (0.382 − 0.923i)7-s + (6.59 − 6.59i)8-s + (2.22 − 2.22i)9-s + (−2.56 + 6.19i)10-s + (−1.58 − 0.655i)11-s + (5.14 + 12.4i)12-s + 0.206i·13-s + (−2.51 + 1.04i)14-s + (−4.31 − 4.31i)15-s − 14.5·16-s + (−1.71 + 3.74i)17-s + ⋯
L(s)  = 1  + (−1.36 − 1.36i)2-s + (1.32 − 0.547i)3-s + 2.71i·4-s + (−0.421 − 1.01i)5-s + (−2.54 − 1.05i)6-s + (0.144 − 0.349i)7-s + (2.32 − 2.32i)8-s + (0.742 − 0.742i)9-s + (−0.811 + 1.96i)10-s + (−0.477 − 0.197i)11-s + (1.48 + 3.58i)12-s + 0.0572i·13-s + (−0.672 + 0.278i)14-s + (−1.11 − 1.11i)15-s − 3.63·16-s + (−0.416 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $-0.763 + 0.646i$
Analytic conductor: \(0.950219\)
Root analytic conductor: \(0.974792\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 119,\ (\ :1/2),\ -0.763 + 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.261825 - 0.714601i\)
\(L(\frac12)\) \(\approx\) \(0.261825 - 0.714601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (1.71 - 3.74i)T \)
good2 \( 1 + (1.92 + 1.92i)T + 2iT^{2} \)
3 \( 1 + (-2.29 + 0.949i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.942 + 2.27i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (1.58 + 0.655i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 0.206iT - 13T^{2} \)
19 \( 1 + (-1.53 - 1.53i)T + 19iT^{2} \)
23 \( 1 + (-7.04 - 2.91i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.91 + 4.62i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3.49 + 1.44i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-6.34 + 2.62i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.16 - 5.23i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.63 - 2.63i)T - 43iT^{2} \)
47 \( 1 + 2.92iT - 47T^{2} \)
53 \( 1 + (-8.11 - 8.11i)T + 53iT^{2} \)
59 \( 1 + (2.46 - 2.46i)T - 59iT^{2} \)
61 \( 1 + (0.668 - 1.61i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 8.94T + 67T^{2} \)
71 \( 1 + (3.72 - 1.54i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.80 - 4.34i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.01 - 1.24i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (9.15 + 9.15i)T + 83iT^{2} \)
89 \( 1 - 1.47iT - 89T^{2} \)
97 \( 1 + (-2.29 - 5.54i)T + (-68.5 + 68.5i)T^{2} \)
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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

−13.07579810047215860416042426941, −12.00058907334959957608755847587, −10.93273940422232681856937557820, −9.682991544074569296556975247063, −8.785278164873219204946694761090, −8.166086243163992449774422990836, −7.42209977125788189144106666780, −4.14751197892725589418106994515, −2.81888475928292667877931736000, −1.30980219393679193198840323420, 2.75452065004112177574187742164, 5.00642419452250988000393904741, 6.78353228945099884930485733148, 7.53707972735361225192137906551, 8.577925770139875791757990698994, 9.276392841320862895894144248062, 10.27536915526660746618266515758, 11.19418652374968404128323047861, 13.58099023819110972934979994820, 14.54267100145147192090158307830

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