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Properties

Label 2-74-37.26-c3-0-8
Degree $2$
Conductor $74$
Sign $-0.696 + 0.717i$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (3.42 − 5.92i)3-s + (−1.99 + 3.46i)4-s + (2.94 − 5.09i)5-s − 13.6·6-s + (8.40 − 14.5i)7-s + 7.99·8-s + (−9.93 − 17.2i)9-s − 11.7·10-s − 26.2·11-s + (13.6 + 23.7i)12-s + (−19.5 + 33.8i)13-s − 33.6·14-s + (−20.1 − 34.8i)15-s + (−8 − 13.8i)16-s + (−11.4 − 19.7i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.658 − 1.14i)3-s + (−0.249 + 0.433i)4-s + (0.263 − 0.455i)5-s − 0.931·6-s + (0.453 − 0.785i)7-s + 0.353·8-s + (−0.368 − 0.637i)9-s − 0.372·10-s − 0.719·11-s + (0.329 + 0.570i)12-s + (−0.416 + 0.722i)13-s − 0.641·14-s + (−0.346 − 0.600i)15-s + (−0.125 − 0.216i)16-s + (−0.162 − 0.282i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 + 0.717i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.696 + 0.717i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ -0.696 + 0.717i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.581395 - 1.37445i\)
\(L(\frac12)\) \(\approx\) \(0.581395 - 1.37445i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
37 \( 1 + (131. - 182. i)T \)
good3 \( 1 + (-3.42 + 5.92i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-2.94 + 5.09i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-8.40 + 14.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + 26.2T + 1.33e3T^{2} \)
13 \( 1 + (19.5 - 33.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (11.4 + 19.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-38.0 + 65.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 40.7T + 1.21e4T^{2} \)
29 \( 1 - 145.T + 2.43e4T^{2} \)
31 \( 1 - 242.T + 2.97e4T^{2} \)
41 \( 1 + (-108. + 188. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 - 372.T + 7.95e4T^{2} \)
47 \( 1 + 83.4T + 1.03e5T^{2} \)
53 \( 1 + (197. + 341. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (72.7 + 126. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (28.8 - 49.8i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (402. - 697. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-91.1 + 157. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 751.T + 3.89e5T^{2} \)
79 \( 1 + (371. - 643. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-512. - 887. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (376. + 652. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 980.T + 9.12e5T^{2} \)
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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

−13.59154335503592470886931503448, −12.69217716066819888394503043401, −11.59872485212586080706234725661, −10.27686937745801698723428003827, −8.961540482918725991713365405447, −7.895856260385713412506376937377, −6.96671132383957255979234582654, −4.71476732805425893519458043292, −2.59321280994575655754726392539, −1.12102056398389703499491159389, 2.78569588061363515472380408982, 4.67654268088849477023687186482, 5.97162332133673193887359805850, 7.82624375577214311676022640396, 8.755694512333946887671417254309, 9.953248874927150888429393778197, 10.58762441944922802796684056639, 12.30626394186391392501721427043, 13.95894918866992435310021523919, 14.72346269683659497459229199884

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