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Properties

Label 2-304-16.13-c1-0-6
Degree $2$
Conductor $304$
Sign $0.487 - 0.873i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
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  + (0.0411 − 1.41i)2-s + (1.66 + 1.66i)3-s + (−1.99 − 0.116i)4-s + (−2.15 + 2.15i)5-s + (2.42 − 2.28i)6-s + 1.65i·7-s + (−0.246 + 2.81i)8-s + 2.54i·9-s + (2.96 + 3.14i)10-s + (−2.90 + 2.90i)11-s + (−3.12 − 3.51i)12-s + (−1.83 − 1.83i)13-s + (2.33 + 0.0680i)14-s − 7.18·15-s + (3.97 + 0.464i)16-s + 6.74·17-s + ⋯
L(s)  = 1  + (0.0290 − 0.999i)2-s + (0.960 + 0.960i)3-s + (−0.998 − 0.0581i)4-s + (−0.965 + 0.965i)5-s + (0.988 − 0.932i)6-s + 0.624i·7-s + (−0.0871 + 0.996i)8-s + 0.846i·9-s + (0.936 + 0.993i)10-s + (−0.877 + 0.877i)11-s + (−0.903 − 1.01i)12-s + (−0.508 − 0.508i)13-s + (0.624 + 0.0181i)14-s − 1.85·15-s + (0.993 + 0.116i)16-s + 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.487 - 0.873i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04902 + 0.615841i\)
\(L(\frac12)\) \(\approx\) \(1.04902 + 0.615841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0411 + 1.41i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.66 - 1.66i)T + 3iT^{2} \)
5 \( 1 + (2.15 - 2.15i)T - 5iT^{2} \)
7 \( 1 - 1.65iT - 7T^{2} \)
11 \( 1 + (2.90 - 2.90i)T - 11iT^{2} \)
13 \( 1 + (1.83 + 1.83i)T + 13iT^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
23 \( 1 - 2.23iT - 23T^{2} \)
29 \( 1 + (3.73 + 3.73i)T + 29iT^{2} \)
31 \( 1 - 6.94T + 31T^{2} \)
37 \( 1 + (1.89 - 1.89i)T - 37iT^{2} \)
41 \( 1 - 6.17iT - 41T^{2} \)
43 \( 1 + (-4.55 + 4.55i)T - 43iT^{2} \)
47 \( 1 - 8.44T + 47T^{2} \)
53 \( 1 + (4.47 - 4.47i)T - 53iT^{2} \)
59 \( 1 + (-1.79 + 1.79i)T - 59iT^{2} \)
61 \( 1 + (7.56 + 7.56i)T + 61iT^{2} \)
67 \( 1 + (4.88 + 4.88i)T + 67iT^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + 4.06iT - 73T^{2} \)
79 \( 1 + 7.88T + 79T^{2} \)
83 \( 1 + (-7.61 - 7.61i)T + 83iT^{2} \)
89 \( 1 - 9.46iT - 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−11.90064876341123723374054870108, −10.72533510123381191857511070793, −10.04245926769422756662947104021, −9.440326636676119705722580646723, −8.123012268474463974287805697894, −7.61122153893411796574970085255, −5.46801093848939093796867774874, −4.28773843328169822550788902021, −3.24817718867667757972830705154, −2.61246707595077421810080350924, 0.842679506199536630289273118868, 3.22479195962462524725587545233, 4.45984423908339490989377226765, 5.64513988491314006285646613912, 7.20396380229519585063243776387, 7.69599527454050078885378495283, 8.360012579014093229338196842957, 9.108970257918487329128948736917, 10.44472964514933051877631756014, 12.08463767962222982788081407593

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