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Properties

Label 2-425-17.8-c1-0-2
Degree $2$
Conductor $425$
Sign $-0.797 + 0.602i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + (−1.91 + 1.91i)2-s + (−0.405 + 0.977i)3-s − 5.34i·4-s + (−1.09 − 2.65i)6-s + (−1.31 + 0.544i)7-s + (6.41 + 6.41i)8-s + (1.32 + 1.32i)9-s + (1.82 + 4.41i)11-s + (5.23 + 2.16i)12-s − 1.14i·13-s + (1.47 − 3.56i)14-s − 13.9·16-s + (0.876 − 4.02i)17-s − 5.09·18-s + (−5.07 + 5.07i)19-s + ⋯
L(s)  = 1  + (−1.35 + 1.35i)2-s + (−0.233 + 0.564i)3-s − 2.67i·4-s + (−0.448 − 1.08i)6-s + (−0.496 + 0.205i)7-s + (2.26 + 2.26i)8-s + (0.442 + 0.442i)9-s + (0.551 + 1.33i)11-s + (1.50 + 0.625i)12-s − 0.317i·13-s + (0.394 − 0.951i)14-s − 3.47·16-s + (0.212 − 0.977i)17-s − 1.20·18-s + (−1.16 + 1.16i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.797 + 0.602i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.797 + 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132808 - 0.396064i\)
\(L(\frac12)\) \(\approx\) \(0.132808 - 0.396064i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (-0.876 + 4.02i)T \)
good2 \( 1 + (1.91 - 1.91i)T - 2iT^{2} \)
3 \( 1 + (0.405 - 0.977i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (1.31 - 0.544i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.82 - 4.41i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 1.14iT - 13T^{2} \)
19 \( 1 + (5.07 - 5.07i)T - 19iT^{2} \)
23 \( 1 + (-1.83 - 4.43i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (7.95 + 3.29i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.22 + 2.96i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (0.968 - 2.33i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-5.63 + 2.33i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (3.05 + 3.05i)T + 43iT^{2} \)
47 \( 1 - 5.03iT - 47T^{2} \)
53 \( 1 + (3.89 - 3.89i)T - 53iT^{2} \)
59 \( 1 + (0.928 + 0.928i)T + 59iT^{2} \)
61 \( 1 + (6.05 - 2.50i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 1.95T + 67T^{2} \)
71 \( 1 + (-0.794 + 1.91i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-3.27 - 1.35i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.477 + 1.15i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-2.88 + 2.88i)T - 83iT^{2} \)
89 \( 1 - 8.34iT - 89T^{2} \)
97 \( 1 + (2.47 + 1.02i)T + (68.5 + 68.5i)T^{2} \)
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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

−11.23355984830446391566215186511, −10.32080047547573928170236079963, −9.618463366503284492459662648106, −9.250437578671221881624354718862, −7.85119415119809878611875730559, −7.32917693934420598925575201345, −6.28076217457057348774286872662, −5.37607659738694909535047289787, −4.30894600886466414911326133273, −1.79227516361812237757506775113, 0.43732868014236505987587940009, 1.71931368818032971335587268716, 3.18350220406112610422471209417, 4.10773666771761788846428000736, 6.35706916992920453058249320518, 7.06609679217276457452912030621, 8.301636302810538469898371696352, 8.940891515685423021809312336831, 9.736103744164775516928088514792, 10.85459977987106961802786586608

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